Apparatus and method for predicting vertical stress fields

ABSTRACT

A method of estimating at least one of stress and pore fluid pressure in an earth formation is disclosed. The method includes: discretizing a domain including at least a portion of the earth formation into a plurality of cells, each cell including a respective density value; dividing the domain into a first region and a second region, the first region including a surface of the earth formation; vertically integrating the respective density values in the first region; and estimating the total vertical stress for each cell in the first region and the second region by estimating a point load based on the respective density value.

BACKGROUND

Determination of pore fluid pressure is an important aspect ofsubterranean drilling, exploration and completion operations.Determination of pore fluid pressure is important in maintaining properfluid pressures to maximize the effectiveness of drilling, production orother operations. For example, the drilling fluid pressure applied bydrilling fluid pumped downhole through a drillstring must be sufficientto control hydrostatic pressure in a wellbore to prevent blowouts andmaintain optimum drilling rates.

Typically, the pore fluid pressure at a point in a formation has beencalculated by considering a difference between total vertical andeffective vertical stress at the point of interest. Conventionally,total vertical stress is estimated by vertical integration of densitydata. On the other hand, there are different approaches for estimationof effective vertical stresses.

Total vertical stress distribution in the Earth may be affected by manyfactors including surface topology and density heterogeneities. Theeffect of these factors on total vertical stresses decays with depthbelow the surface or below the heterogeneity. For example, totalvertical stresses are significantly affected by topology close to thesurface, but with increasing depth, they approach the stressdistribution for a horizontal surface at average elevation.

Conventionally used vertical integration of density implicitly assumesthat the gravitational load of an infinitesimal rock element iscompletely transferred to the element below it. As a result of thisassumption, the influence of the gravitational load of an element on thevertical stress distribution does not decay with depth but istransferred to all the elements below it. Depending on the surfacetopography and density distribution, this assumption can result inoverestimation or underestimation of the total vertical stresses and inturn, overestimation or underestimation of the formation pore pressuresderived from the total vertical stresses.

BRIEF DESCRIPTION

Disclosed herein is a method of estimating at least one of stress andpore fluid pressure in an earth formation, including: discretizing adomain including at least a portion of the earth formation into aplurality of cells, each cell including a respective density value;dividing the domain into a first region and a second region, the firstregion including a surface of the earth formation; verticallyintegrating the respective density values in the first region; andestimating the total vertical stress for each cell in the first regionand the second region by estimating a point load based on the respectivedensity value.

Also disclosed herein is a system for estimating at least one of stressand pore fluid pressure in an earth formation, including: a downholetool configured to be disposed in a borehole in the earth formation; atleast one sensor associated with the downhole tool configured togenerate data relating to a density of the earth formation; a processorin operable communication with the sensor, for receiving the data, theprocessor performing: discretizing a domain including at least a portionof an earth formation into a plurality of cells, each cell including arespective density value; dividing the domain into a first region and asecond region, the first region including a surface of the earthformation; vertically integrating the respective density values in thefirst region; and estimating the total vertical stress for each cell inthe first region and the second region by estimating a point load basedon the respective density value.

Further disclosed herein is a computer program product stored on machinereadable media for estimating at least one of stress and pore fluidpressure in an earth formation by executing machine implementedinstructions, the instructions for: discretizing a domain including atleast a portion of the earth formation into a plurality of cells, eachcell including a respective density value; dividing the domain into afirst region and a second region, the first region including a surfaceof the earth formation; vertically integrating the respective densityvalues in the first region; and estimating the total vertical stress foreach cell in the first region and the second region by estimating apoint load based on the respective density value.

BRIEF DESCRIPTION OF THE DRAWINGS

The following descriptions should not be considered limiting in any way.With reference to the accompanying drawings, like elements are numberedalike:

FIG. 1 depicts an exemplary embodiment of a well drilling, productionand/or logging system;

FIG. 2 depicts a flow chart providing an exemplary method of predictinga force such as vertical stress and/or formation pore fluid pressure inan earth formation;

FIG. 3 depicts a cross-sectional view of a domain including an earthformation and an associated density data matrix;

FIG. 4 depicts a cross-sectional view of a domain and a cross-sectionalrepresentation of an associated distributed surface load;

FIG. 5 depicts density data matrices of the domain of FIG. 4;

FIG. 6 depicts a cell of a two-dimensional density data matrix and anassociated idealized point load; and

FIG. 7 depicts a cell of a three-dimensional density data matrix and anassociated idealized point load;

DETAILED DESCRIPTION

A detailed description of one or more embodiments of the disclosedapparatus and method are presented herein by way of exemplification andnot limitation with reference to the Figures.

Referring to FIG. 1, an exemplary embodiment of a portion of a welldrilling, production and/or logging system 10 includes a conduit orstring 12, such as a drillstring or production string. The string 12 isconfigured to be disposed in a borehole 14 for performing operationssuch as drilling the borehole 14, making measurements of properties ofthe borehole 14 and/or the surrounding formation downhole, andfacilitating hydrocarbon production. As a matter of convention, a depthof the borehole 14 is described along a z-axis, while a cross-section isprovided on a plane described by an x-axis and a y-axis.

In one example, the drill string 12 includes lengths of drill pipe ordrill segments 16 which drive a drill bit 18. Drilling fluid 20 ispumped or otherwise flows through the drill string 12 toward the drillbit 18, and exits into the borehole 14. The drilling fluid 20 (alsoreferred to as “drilling mud”) generally includes a mixture of liquidssuch as water, drilling fluid, mud, oil, gases, and formation fluids asmay be indigenous to the surroundings.

The string 12 may include equipment therein such as a logging instrumentor logging tool 22 for performing various measurements of the borehole,downhole components and/or the formation. In one embodiment, the loggingtool 22 is configured as a “measurement while drilling” (MWD) or“logging while drilling” (LWD) tool. In another embodiment, the loggingtool 22 is configured to be lowered into the borehole 14 after drilling,such as by a cable or wireline. Exemplary tools 22 include sensors forgenerating data such as resistivity, density, gamma ray, pressure,strain and stress data. In one embodiment, the tool 22 is configured tocollect and/or process data for predicting or estimating a verticalstress field and/or pore fluid pressures of the formation.

The logging tool 22 includes at least one sensor 24 for sensing variouscharacteristics of the borehole 14, the formation and/or downholecomponents. In one embodiment, the at least one sensor 24 is incommunication with downhole electronics 26 that may receive input fromthe sensor 24 and provide for at least one of operational control anddata analysis. The downhole electronics 26 may include, withoutlimitation, a power supply, a transformer, a battery, a processor,memory, storage, at least one communications interface and the like.

In one embodiment, the logging tool 22, sensor 24 and/or electronics 26are operably coupled in communication with surface equipment 28. Thesurface equipment 28 may provide power to the tool 22 and/or otherdownhole components, as well as provide computing and processingcapabilities for at least one of control of operations and analysis ofdata. A communications channel is included for communication with thesurface equipment 28, and may operate via pulsed mud, wired pipe, andother technologies as are known in the art.

In one embodiment, the system 10 is operably connected to a downhole orsurface processing unit, such as surface equipment 28, which may act tocontrol various components of the system 10, such as drilling, loggingand production components or subs. Other components include machinery toraise or lower segments and to operably couple segments, andtransmission devices. The downhole or surface processing unit may alsocollect and process data generated by the system 10 during drilling,production or other operations.

FIG. 2 illustrates a method 40 of predicting or estimating a force in anearth formation. Such a force includes stress and pressure, such as thevertical stress field and/or formation pore fluid pressures in an earthformation. Prediction of the vertical stress field and/or formation porefluid pressures includes estimating the total vertical stress of theformation. In this method, estimation of total vertical stress includesa calculation approach utilizing the Boussinesq's solution for a pointload on a half space.

The method 40 includes one or more stages 41-49. The method 40 isdescribed herein in conjunction with the system 10, although the method40 may be performed in conjunction with any number and configuration ofsensors, tools, processors or other machinery. The method 40 may beutilized as a workflow or as part of one or more workflows, such asvertical stress, pore fluid pressure and horizontal stress estimationworkflows. In one embodiment, the method 40 includes the execution ofall of stages 41-49 in the order described. However, certain stages maybe omitted, stages may be added, or the order of the stages changed.

In the first stage 41, density data is calculated or measured at variousdepths (z-axis locations). In one embodiment, data is collected at eachselected depth at a plurality of x-axis and/or y-axis locations. Theplurality of x-axis and/or y-axis locations may correspond to sensorlocations in one or more boreholes 14 and/or in a sensor array disposedwith the tool 22. In one embodiment, density data is estimated from datagenerated by one or more gamma ray detectors.

In the second stage 42, referring to FIG. 3, a region or domain 50 isselected that includes locations at which density data has beengenerated. In one embodiment, the domain 50 is a two-dimensional plane.In another embodiment, the domain 50 is a three-dimensional region. Forconvention purposes, the domain 50 has a z-axis corresponding to a depthof the location, and an x-axis and/or y-axis that is orthogonal to thez-axis. In one embodiment, the domain 50 includes a surface region 52including a surface topology 54 and a subterranean region 56.

In the third stage 43, the domain 50 is discretized into a plurality ofcells forming a matrix. In one embodiment, the domain is two-dimensionaland the cells are rectangular cells having dimensions referred to as“Δx” and “Δz”. The matrix includes a number of rows “M” in thez-direction and a number of columns “N” in the x-direction.

In another embodiment, the domain 50 is three-dimensional and the cellsare rectangular prism cells having dimensions “Δx”, “Δy”, and “Δz”. Inthis embodiment, the matrix includes a number of rows “M” in thez-direction, a number of columns “N” in the x-direction and a number ofcolumns “R” in the y-direction.

In the fourth stage 44, the cells are populated with density data. Thedomain is partitioned into two regions 58, 60 separated by a “halfspace” 62. For two-dimensional domains, the half space 62 is a lineextending along the x-axis. For three-dimensional domains, the halfspace 62 is a plane extending along the x- and y-axes. In oneembodiment, the half space 62 is positioned at a selected depth relativeto the surface topology 54. For example, the half space 62 is positionedat a depth corresponding to a lowest depth of the surface topology 54.

In one embodiment, for a two-dimensional domain, each cell thus includesa density value ρ_(i,j), where “i” is a row number 1 through M and “j”is a column number 1 through N. The domain is sectioned into two regions58 and 60, which are bounded by the half space line 62. A first region58 includes rows above the half space 62, shown as rows 1 through p, anda second region 60 includes rows below the half space 62, shown as rowsr through M.

In another embodiment, for a three-dimensional domain, each cellincludes a density value ρ_(i,j,k), where “i” is a row number 1 throughM, “j” is an x-axis column number 1 through N and “k” is a y-axis columnnumber 1 through R. The two regions 58, 60 are bounded by a half spaceplane 62, the first region 58 including rows 1 through p and the secondregion 60 including rows r through M.

In the fifth stage 45, referring to FIGS. 4 and 5, the total verticalstress (i.e., overburden stress) in each column in the first region 58(i.e., above the half space 62) is estimated or calculated, for example,by using vertical integration of the density data. The total verticalstress data is stored and the total vertical stress corresponding to thedepth of the half space 62 is applied on the second region 60 as adistributed surface load 64.

Implementation of this stage is shown in FIG. 5. A new discrete domain66 is created that includes rows p and r through M, and density valuesin the original domain 50 are transferred to the new discrete domain 66.Density values in the row above depth to half space (row p) are replacedby equivalent density values, ρ*.

In one embodiment, where the domain 50, 66 is two-dimensional, ρ* isrepresented as “ρ_(i)*”, where:)

ρ_(i) *=S _(p,i) /Δz, i=1 . . . N

“S_(p,i)” is the total vertical stress value calculated by verticalintegration for the first region in each cell (p,i) located in the rowp, “Δz” is the vertical dimension of the cell, and N is the number ofelements “i” in the x-direction.

In another embodiment, where the domain 50, 66 is three-dimensional, ρ*is represented as “ρ_(i,j)*”, where:

ρ_(i,j) *S= _(p,i,j) /Δz, i=1 . . . N, j=1 . . . R

“S_(p,i,j)” is the total vertical stress value calculated by verticalintegration for the first region 58 in each cell (p,i,j), and “N” and“R” are the number elements “i” and “j” in the x- and y-directions,respectively.

In the sixth stage 46, the gravitational load in each cell is idealizedas a point load acting at the bottom center of the cell. The point loadlocation in each cell is exemplary, as other locations such as a centeror top center location can be used.

In one embodiment, shown in FIG. 6, the domain 66 is a two-dimensionaldomain having a plurality of two-dimensional (such as rectangular) cells68. In this embodiment, the cell numbers “i” and “k” are the coordinatesof each cell 68 in the x-direction and z-direction, respectively, andthe density associated with the cell 68 is denoted as “ρ_(k,i)”. Theinduced total vertical stress, Δσ_(ν), at each cell 68 is calculated fora selected point load location 70, designated (x_(i), z_(k)). In oneembodiment, the total vertical stress Δσ_(ν) is calculated for aplurality of points (x_(m), z_(l)) relative to the selected point(x_(i), z_(k)) as:

${\Delta \; \sigma_{{{(\upsilon)}m},l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{{\rho_{k,i}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = x_{m}}} \\{\frac{2P_{k,i}}{\pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; z^{2}}} \right)^{2}}} & {z_{l} > z_{k}}\end{Bmatrix}$

where “Δσ_((ν)m,i)” is the induced vertical stress at a point (x_(m),z_(l)), x_(m) and z_(l) are the x- and z-coordinates of each of theplurality of points relative to the cell 68, and x_(i) and z_(k) are thex- and z-coordinates of the location of the idealized point load 70. δzis equal to the difference between z_(l) and z_(k), and δx is equal tothe difference between x_(m) and x_(i). P_(k,i) is a vertical point loadvalue at the idealized point and may be calculated as:

P _(k,i)=ρ_(k,i) ·Δx·Δz.

Δx is a dimension of the cell along the x-axis, and Δz is a dimension ofthe cell along the z-axis.

In another embodiment, shown in FIG. 7, the domain 66 is athree-dimensional domain. In this embodiment, the cell numbers “i”, “j”and “k” are the numbers in the x-direction, y-direction and z-direction,respectively, and the density associated with the cell 68 is denoted as“ρ_(k,i,j)”. The induced total vertical stress, Δσ_(ν), at each cell 68is calculated for a selected point load location 70, designated (x_(i),y_(i), z_(k)). In one embodiment, the total vertical stress Δσ_(ν) iscalculated for a plurality of points (x_(m), y_(n), z_(l)) relative tothe selected point (x_(i), y_(j), z_(k)) as:

${\Delta \; \sigma_{{{(\upsilon)}m},n,l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} y_{n}} \neq y_{j}}} \\{{\rho_{k,i,j}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = {{{x_{m}\&}\mspace{14mu} y_{n}} = y_{j}}}} \\{\frac{3P_{k,i,j}}{2\; \pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; y^{2}} + {\delta \; z^{2}}} \right)^{\frac{5}{2}}}} & {z_{l} > z_{k}}\end{Bmatrix}$

where Δσ_((ν)m,n,l) is the induced vertical stress at a point (x_(m),y_(n), z_(l)), x_(m), y_(n) and z_(l) are the respective x-, y- andz-coordinates of each of the plurality of points relative to the cell68, and x_(i), y_(j) and z_(k) are the x-, y- and z-coordinates of thelocation of the idealized point load 70. δz is equal to the differencebetween z_(l) and z_(k), δx is equal to the difference between x_(m) andx_(i), and δy is equal to the difference between y_(n) and y_(j).P_(k,i,j) is a vertical point load value at the idealized point 70 andmay be calculated as:

P _(k,i,j)=ρ_(k,i,j) ·Δx·Δy·Δz.

Δx is a dimension of the cell along the x-axis, Δy is a dimension of thecell along the y-axis, and Δz is a dimension of the cell along thez-axis.

In the seventh stage 47, the total vertical stress σ_(ν) at each cell 68is calculated by summing all of the induced vertical stressesΔσ_((ν)m,n,l). The total vertical stress σ_(ν) for each cell 68 in thefirst region 58 is merged with the total vertical stress σ_(ν) for eachcell in the second region 60 to form a complete total vertical stressfield of the domain 50.

In the eighth stage 48, effective vertical stress is estimated orcalculated based on any suitable method or technique. For example, theeffective vertical stress is calculated for a selected cell 68 frominterval velocities based on empirical relationships that are calibratedagainst well-based data.

In the ninth stage 49, the pore fluid pressure is estimated bysubtracting the effective vertical stress from the total verticalstress. In one embodiment, the effective vertical stress for a selectedcell 68 is subtracted from the total vertical stress σ_(ν) for theselected cell 68.

In addition, various other properties of the formation can be estimatedusing the vertical stress calculations described herein. For example,the estimated total vertical stress and pore fluid pressure are used toestimate horizontal stresses. Total maximum and total minimum horizontalstresses can be calculated as:

ESR(min)=(shmin−Pp)/(Sv−Pp),

ESR(max)=(sHmax−Pp)/(Sv−Pp),

where ESR(min) is the effective stress ratio for minimum horizontalstress, ESR(max) is the effective stress ratio for maximum horizontalstress, Pp is the pore fluid pressure, Shmin is the total minimumhorizontal stress, sHmax is the total maximum horizontal stress, and Svis the total vertical stress.

As described herein, “drillstring” or “string” refers to any structureor carrier suitable for lowering a tool through a borehole or connectinga drill bit to the surface, and is not limited to the structure andconfiguration described herein. For example, the string 12 is configuredas a hydrocarbon production string or formation evaluation string. Theterm “carrier” as used herein means any device, device component,combination of devices, media and/or member that may be used to convey,house, support or otherwise facilitate the use of another device, devicecomponent, combination of devices, media and/or member. Exemplarynon-limiting carriers include drill strings of the coiled tube type, ofthe jointed pipe type and any combination or portion thereof. Othercarrier examples include casing pipes, wirelines, wireline sondes,slickline sondes, drop shots, downhole subs, BHA's and drill strings.

In support of the teachings herein, various analyses and/or analyticalcomponents may be used, including digital and/or analog systems. Thesystem may have components such as a processor, storage media, memory,input, output, communications link (wired, wireless, pulsed mud, opticalor other), user interfaces, software programs, signal processors(digital or analog) and other such components (such as resistors,capacitors, inductors and others) to provide for operation and analysesof the apparatus and methods disclosed herein in any of several mannerswell-appreciated in the art. It is considered that these teachings maybe, but need not be, implemented in conjunction with a set of computerexecutable instructions stored on a computer readable medium, includingmemory (ROMs, RAMs), optical (CD-ROMs), or magnetic (disks, harddrives), or any other type that when executed causes a computer toimplement the method of the present invention. These instructions mayprovide for equipment operation, control, data collection and analysisand other functions deemed relevant by a system designer, owner, user orother such personnel, in addition to the functions described in thisdisclosure.

The apparatuses and methods described herein provide various advantagesover existing methods and devices, in that the apparatuses and methodsproduce described herein result in superior pore fluid pressurepredictions and improved calculation methods as compared to prior arttechniques.

The pore pressure prediction methods described herein provide animprovement in workflows pertaining to calculation of formationstresses, pressures and other properties. For example, the improvementin the total vertical stress, i.e. overburden stress, calculationprovides an improvement in any related workflow. In addition, most ofthe currently used methods start with calculation of total vertical oroverburden stress. Thus, the methods described herein improve not onlypore pressure prediction workflows but also other workflows such asestimation of total horizontal stresses.

Prior art methods calculate pore fluid pressure and generally estimatetotal vertical stress by vertical integration of density data. Suchintegration cannot capture the decay of the effect of topology andheterogeneities as a function of depth, and thus these prior art methodscan lead to unrealistic pore fluid pressure predictions and unrealisticinput for borehole stability predictions during drilling and production.In contrast, the apparatuses and methods described herein produceresults that reflect the decay with depth of the total vertical stress,and thus produce more accurate and realistic results.

One skilled in the art will recognize that the various components ortechnologies may provide certain necessary or beneficial functionalityor features. Accordingly, these functions and features as may be neededin support of the appended claims and variations thereof, are recognizedas being inherently included as a part of the teachings herein and apart of the invention disclosed.

While the invention has been described with reference to exemplaryembodiments, it will be understood by those skilled in the art thatvarious changes may be made and equivalents may be substituted forelements thereof without departing from the scope of the invention. Inaddition, many modifications will be appreciated by those skilled in theart to adapt a particular instrument, situation or material to theteachings of the invention without departing from the essential scopethereof. Therefore, it is intended that the invention not be limited tothe particular embodiment disclosed as the best mode contemplated forcarrying out this invention, but that the invention will include allembodiments falling within the scope of the appended claims.

1. A method of estimating stress in an earth formation, comprising:dividing a domain including at least a portion of an earth formationinto a first region and a second region; estimating a first verticalstress in the first region and representing the first vertical stress asat least one point load; and estimating a second vertical stress in thesecond region by a point load based method using the first verticalstress.
 2. The method of claim 1, further comprising discretizing thedomain into a plurality of cells, each cell including a respectivedensity value being representative of a selected location in the domain.3. The method of claim 1, further comprising estimating at least one offorce, pressure, pore pressure, total vertical stress, and effectivevertical stress based on the second vertical stress.
 4. The method ofclaim 1, wherein the first region and the second region are divided by ahalf space that is positioned at a selected depth relative to a topologyof the at least a portion of the earth formation.
 5. The method of claim2, wherein estimating the second vertical stress includes estimating apoint load for the second vertical stress in each cell in the secondregion.
 6. The method of claim 2, wherein the domain is two-dimensionaland includes a z-axis corresponding to a depth of the earth formationand an x-axis orthogonal to the z-axis, and a vertical stress in eachcell is calculated based on the following equation:${\Delta \; \sigma_{{{(\upsilon)}m},l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{{\rho_{k,i}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = x_{m}}} \\{\frac{2P_{k,i}}{\pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; z^{2}}} \right)^{2}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,l) is an induced vertical stress at apoint (x_(m), z_(l)) in the cell, x_(m) and z_(l) are coordinates of atleast one location on the x-axis and the z-axis respectively, x_(i) andz_(k) are coordinates of the point load on the x-axis and the z-axisrespectively, ρ_(k,i) is the respective density value, δz is equal tothe difference between z_(l) and z_(k), δx is equal to the differencebetween x_(m) and x_(i), P_(k,i) is a vertical point load valuerepresented by:P _(k,i)=ρ_(k,i) ·Δx·Δz, Δx is a dimension of the cell along the x-axis,and Δz is a dimension of the cell along the z-axis.
 7. The method ofclaim 2, wherein the domain is three-dimensional and includes a z-axiscorresponding to a depth of the earth formation, an x-axis orthogonal tothe z-axis, and a y-axis orthogonal to the x-axis and the z-axis, and avertical stress in each cell is calculated based on the followingequation: ${\Delta \; \sigma_{{{(\upsilon)}m},n,l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} y_{n}} \neq y_{j}}} \\{{\rho_{k,i,j}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = {{{x_{m}\&}\mspace{14mu} y_{n}} = y_{j}}}} \\{\frac{3P_{k,i,j}}{2\; \pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; y^{2}} + {\delta \; z^{2}}} \right)^{\frac{5}{2}}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,mn,l) is an induced vertical stress at apoint (x_(m), y_(n), z_(l)) in the cell, x_(m), y_(r), and z_(l) arecoordinates of at least one location on the x-axis, the y-axis and thez-axis respectively, x_(i), y_(j) and z_(k) are coordinates of the pointload on the x-axis, the y-axis and the z-axis respectively, ρ_(k,i,j) isthe respective density value, δz is equal to the difference betweenz_(l) and z_(k), δx is equal to the difference between x_(m) and x_(i),δy is equal to the difference between y_(n) and y_(j), P_(k,i,j) is avertical point load value represented by:P _(k,i,j)=ρ_(k,i,j) ·Δx·Δy·Δz, Δx is a dimension of the cell along thex-axis, Δy is a dimension of the cell along the y-axis, and Δz is adimension of the cell along the z-axis.
 8. The method of claim 1,wherein the first region includes a topology of the at least a portionof the earth formation.
 9. The method of claim 3, further comprisingestimating a pore fluid pressure based on the effective vertical stressand the total vertical stress.
 10. A system for estimating stress in anearth formation, the system comprising: a tool configured to at leastone of generate and receive density information for the earth formation,the tool configured to perform: dividing a domain including at least aportion of an earth formation into a first region and a second region;estimating a first vertical stress in the first region based on thedensity information and representing the first vertical stress as atleast one point load; and estimating a second vertical stress in thesecond region by a point load based method using the first verticalstress.
 11. The system of claim 10, wherein the first region and thesecond region are divided by a half space that is positioned at aselected depth relative to a topology of the at least a portion of theearth formation.
 12. The system of claim 22, wherein the domain istwo-dimensional and includes a z-axis corresponding to a depth of theearth formation and an x-axis orthogonal to the z-axis, and a verticalstress in each cell is calculated based on the following equation:${\Delta \; \sigma_{{{(\upsilon)}m},l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{{\rho_{k,i}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = x_{m}}} \\{\frac{2P_{k,i}}{\pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; z^{2}}} \right)^{2}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,l) is an induced vertical stress at apoint (x_(m), z_(l)) in the cell, x_(m) and z_(l) are coordinates of atleast one location on the x-axis and the z-axis respectively, x_(i) andz_(k) are coordinates of the point load on the x-axis and the z-axisrespectively, ρ_(k,i) is the respective density value, δz is equal tothe difference between z_(l) and z_(k), δx is equal to the differencebetween x_(m) and x_(i), P_(k,i) is a vertical point load valuerepresented by:P _(k,i)=ρ_(k,i) ·Δx·Δz, Δx is a dimension of the cell along the x-axis,and Δz is a dimension of the cell along the z-axis.
 13. The system ofclaim 22, wherein the domain is three-dimensional and includes a z-axiscorresponding to a depth of the earth formation, an x-axis orthogonal tothe z-axis, and a y-axis orthogonal to the x-axis and the z-axis, and avertical stress in each cell is calculated based on the followingequation: ${\Delta \; \sigma_{{{(\upsilon)}m},n,l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} y_{n}} \neq y_{j}}} \\{{\rho_{k,i,j}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = {{{x_{m}\&}\mspace{14mu} y_{n}} = y_{j}}}} \\{\frac{3P_{k,i,j}}{2\; \pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; y^{2}} + {\delta \; z^{2}}} \right)^{\frac{5}{2}}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,n,l) is an induced vertical stress at apoint (x_(m), y_(n), z_(l)) in the cell, x_(m), y_(n), and z_(l) arecoordinates of at least one location on the x-axis, the y-axis and thez-axis respectively, x_(i), y_(j) and z_(k) are coordinates of the pointload on the x-axis, the y-axis and the z-axis respectively, ρ_(k,i,j) isthe respective density value, δz is equal to the difference betweenz_(l) and z_(k), δx is equal to the difference between x_(m) and x_(i),δy is equal to the difference between y_(n) and y_(j), P_(k,i,j) is avertical point load value represented by:P _(k,i,j)ρ_(k,i,j) ·Δx·Δy·Δz, Δx is a dimension of the cell along thex-axis, Δy is a dimension of the cell along the y-axis, and Δz is adimension of the cell along the z-axis.
 14. The system of claim 10wherein the first region includes a topology of the at least a portionof the earth formation.
 15. The system of claim 14 wherein the tool isconfigured to further perform estimating at least one of force,pressure, pore pressure, effective vertical stress and, total verticalstress based on the second vertical stress.
 16. A computer programproduct stored on machine readable media for estimating at least one ofstress and pore fluid pressure in an earth formation by executingmachine implemented instructions, the instructions for: dividing adomain including at least a portion of an earth formation into a firstregion and a second region; estimating a first vertical stress in thefirst region and representing the first vertical stress as at least onepoint load; and estimating a second vertical stress in the second regionby a point load based method using the first vertical stress.
 17. Thecomputer program product of claim 16, wherein the first region and thesecond region are divided by a half space that is positioned at aselected depth relative to a topology of the at least a portion of theearth formation.
 18. The computer program product of claim 24, whereinestimating the second vertical stress includes estimating a point loadfor a vertical stress in each cell in the second region.
 19. Thecomputer program product of claim 24, wherein the domain istwo-dimensional and includes a z-axis corresponding to a depth of theearth formation and an x-axis orthogonal to the z-axis, and a verticalstress in each cell is calculated based on the following equation:${\Delta \; \sigma_{{{(\upsilon)}m},l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{{\rho_{k,i}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = x_{m}}} \\{\frac{2P_{k,i}}{\pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; z^{2}}} \right)^{2}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,l) is an induced vertical stress at apoint (x_(m), z_(l)) in the cell, x_(m) and z_(l) are coordinates of atleast one location on the x-axis and the z-axis respectively, x_(i) andz_(k) are coordinates of the point load on the x-axis and the z-axisrespectively, ρ_(k,i) is the respective density value, δz is equal tothe difference between z_(l) and z_(k), δx is equal to the differencebetween x_(m) and x_(i), P_(k,i) is a vertical point load valuerepresented by:P _(k,i)=ρ_(k,i) ·Δx·Δz, Δx is a dimension of the cell along the x-axis,and Δz is a dimension of the cell along the z-axis.
 20. The computerprogram product of claim 24, wherein the domain is three-dimensional andincludes a z-axis corresponding to a depth of the earth formation, anx-axis orthogonal to the z-axis, and a y-axis orthogonal to the x-axisand the z-axis, and a vertical stress in each cell is calculated basedon the following equation:${\Delta \; \sigma_{{{(\upsilon)}m},n,l}} = \begin{Bmatrix}{0,} & {z_{k} > z_{l}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} \neq x_{m}}} \\{0,} & {z_{k} = {{{z_{l}\&}\mspace{14mu} y_{n}} \neq y_{j}}} \\{{\rho_{k,i,j}\Delta \; z},} & {z_{k} = {{{z_{l}\&}\mspace{14mu} x_{i}} = {{{x_{m}\&}\mspace{14mu} y_{n}} = y_{j}}}} \\{\frac{3P_{k,i,j}}{2\; \pi} \cdot \frac{\delta \; z^{3}}{\left( {{\delta \; x^{2}} + {\delta \; y^{2}} + {\delta \; z^{2}}} \right)^{\frac{5}{2}}}} & {z_{l} > z_{k}}\end{Bmatrix}$ wherein Δσ_((ν)m,n,l) is an induced vertical stress at apoint (x_(m), y_(n), z_(l)) in the cell, x_(m), y_(n), and z_(l) arecoordinates of at least one location on the x-axis, the y-axis and thez-axis respectively, x_(l), y_(j) and z_(k) are coordinates of the pointload on the x-axis, the y-axis and the z-axis respectively, ρ_(k,i,j) isthe respective density value, δz is equal to the difference betweenz_(l) and z_(k), δx is equal to the difference between x_(m) and x_(i),δy is equal to the difference between y_(n) and y_(j), P_(k,i,j) is avertical point load value represented by:P _(k,i,j)=ρ_(k,i,j) ·Δx·Δy·Δz, Δx is a dimension of the cell along thex-axis, Δy is a dimension of the cell along the y-axis, and Δz is adimension of the cell along the z-axis.
 21. The method of claim 2,wherein estimating the first vertical stress includes verticallyintegrating the respective density values in the first region.
 22. Thesystem of claim 10, wherein the tool is configured to further performdiscretizing the domain into a plurality of cells, each cell including arespective density value being representative of a selected location inthe domain.
 23. The system of claim 22, wherein estimating the firstvertical stress includes vertically integrating the respective densityvalues in the first region.
 24. The computer program product of claim16, wherein the instructions are for discretizing the domain into aplurality of cells, each cell including a respective density value beingrepresentative of a selected location in the domain.
 25. The computerprogram product of claim 24, wherein estimating the first verticalstress includes vertically integrating the respective density values inthe first region.